# Multivariate normal distribution

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Notation Probability density function Many sample points from a multivariate normal distribution with ${\boldsymbol {\mu }}=\left[{\begin{smallmatrix}0\\0\end{smallmatrix}}\right]$ and ${\boldsymbol {\Sigma }}=\left[{\begin{smallmatrix}1&3/5\\3/5&2\end{smallmatrix}}\right]$ , shown along with the 3-sigma ellipse, the two marginal distributions, and the two 1-d histograms. ${\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})$ μ ∈ Rk — locationΣ ∈ Rk × k — covariance (positive semi-definite matrix) x ∈ μ + span(Σ) ⊆ Rk $(2\pi )^{-k/2}\det({\boldsymbol {\Sigma }})^{-1/2}\,\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }})^{\!{\mathsf {T}}}{\boldsymbol {\Sigma }}^{-1}(\mathbf {x} -{\boldsymbol {\mu }})\right),$ exists only when Σ is positive-definite μ μ Σ ${\frac {1}{2}}\ln \det \left(2\pi \mathrm {e} {\boldsymbol {\Sigma }}\right)$ $\exp \!{\Big (}{\boldsymbol {\mu }}^{\!{\mathsf {T}}}\mathbf {t} +{\tfrac {1}{2}}\mathbf {t} ^{\!{\mathsf {T}}}{\boldsymbol {\Sigma }}\mathbf {t} {\Big )}$ $\exp \!{\Big (}i{\boldsymbol {\mu }}^{\!{\mathsf {T}}}\mathbf {t} -{\tfrac {1}{2}}\mathbf {t} ^{\!{\mathsf {T}}}{\boldsymbol {\Sigma }}\mathbf {t} {\Big )}$ see below